
theorem Th8: :: generealized YELLOW_3:46, YELLOW10:5
  for L1,L2 being antisymmetric non empty RelStr, D being Subset
  of [:L1,L2:] st ex_sup_of D,[:L1,L2:] holds sup D = [sup proj1 D,sup proj2 D]
proof
  let L1,L2 be antisymmetric non empty RelStr, D be Subset of [:L1,L2:] such
  that
A1: ex_sup_of D,[:L1,L2:];
  per cases;
  suppose
    D <> {};
    hence thesis by A1,YELLOW_3:46;
  end;
  suppose
A2: D = {};
    then ex_sup_of {},L2 by A1,FUNCT_5:8,YELLOW_3:41;
    then
A3: L2 is lower-bounded by Th6;
    ex_sup_of {},L1 by A1,A2,FUNCT_5:8,YELLOW_3:41;
    then L1 is lower-bounded by Th6;
    hence thesis by A1,A3,YELLOW10:5;
  end;
end;
